Spectral Asymptotics for Waveguides with Perturbed Periodic Twisting

Abstract

We consider the twisted waveguide $Ω_θ$, i.e. the domain obtained by the rotation of the bounded cross section $ω\subset {\mathbb R}^{2}$ of the straight tube $Ω: = ω\times {\mathbb R}$ at angle $θ$ which depends on the variable along the axis of $Ω$. We study the spectral properties of the Dirichlet Laplacian in $Ω_θ$, unitarily equivalent under the diffeomorphism $Ω_θ\to Ω$ to the operator $H_{θ'}$, self-adjoint in ${\rm L}^2(Ω)$. We assume that $θ' = β- ε$ where $β$ is a $2π$-periodic function, and $ε$ decays at infinity. Then in the spectrum $σ(H_β)$ of the unperturbed operator $H_β$ there is a semi-bounded gap $(-\infty, {\mathcal E}_0^+)$, and, possibly, a number of bounded open gaps $({\mathcal E}_j^-, {\mathcal E}_j^+)$. Since $ε$ decays at infinity, the essential spectra of $H_β$ and $H_{β- ε}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{β- ε}$ near an arbitrary fixed spectral edge ${\mathcal E}_j^\pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $σ_{\rm disc}(H_{β-ε})$ in a neighbourhood of ${\mathcal E}_j^\pm$. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of $σ_{\rm disc}(H_{β-ε})$ near ${\mathcal E}_j^\pm$ could be represented as a finite orthogonal sum of operators of the form $-μ\frac{d^2}{dx^2} - ηε$, self-adjoint in ${\rm L}^2({\mathbb R})$; here, $μ> 0$ is a constant related to the so-called effective mass, while $η$ is $2π$-periodic function depending on $β$ and $ω$.